1911] HYDRODYNAMTCAL NOTES 7
Consider first an infinite series of simple stationary waves, of which the energy is at one moment wholly potential and [a quarter of] a period later wholly kinetic. If t denote the time and E the total energy, we may write
K.E. = EsirL2nt, RE. = E cos2 nt.
Upon this superpose a similar system, displaced through a quarter wavelength in space and through a quarter period in time. For this, taken by itself, we should have
K.E = E cos2 nt, P.E. = E sin2 nt.
And, the vibrations being conjugate, the potential and kinetic energies of the combined motion may be found by simple addition of the components' and are accordingly independent of the time, and each equal to E. Now the resultant motion is a simple progressive train, of which the potential and kinetic energies are thus seen to be equal.
A similar argument is applicable to prove the equality of energies in the motion of a simple conical pendulum.
It is to be observed that the conclusion is in general limited to vibrations which are infinitely small.
Waves moving into Shallower Water.
The problem proposed is the passage of an infinite train of simple infinitesimal waves from deep water into water which shallows gradually in such a manner that there is no loss of energy by reflexion or otherwise. At any stage the whole energy, being the double of the potential energy, is proportional per unit length to the square of the height; and for motion in two dimensions the only remaining question for our purpose is what are to be regarded as corresponding lengths along the direction of propagation.
In the case of long waves, where the wave-length (X)is long in comparison with the depth (I) of the water, corresponding parts are as the velocities of propagation ( V), or since the periodic time (T) is constant, as X. Conservation of energy then requires that
(height)2 x Inconstant; ........................(1)
or since V varies as $, height varies as l~%*.
But for a dispersive medium corresponding parts are not proportional to V, and the argument requires modification. A uniform regime being established, what we are to equate at two separated places where the waves are of different character is the rate of propagation of energy through these places. It is a general proposition that in any kind of waves the ratio of the energy propagated past a fixed point in unit time to that resident in unit
* Loc. cit. p. 255. , . * 15 ,, ,, For solution read relation.